Knowledge Reasoning Involving Four Types of Syllogisms
Long Wei, Liheng Hao
TL;DR
This work develops a formal framework for knowledge reasoning with four syllogism types built on Square{all} and Square{most}, focusing on the foundational generalized syllogism AMI-1. It proves the validity of AMI-1 and demonstrates that 73 additional valid syllogisms (comprising 19 generalized, 22 generalized modal, 8 classical, and 24 classical modal forms) can be derived from AMI-1 using deductive rules and quantifier reductions, establishing a network of reducible relationships among forms. A knowledge-mining perspective formalizes how these derivations propagate, while discourse reasoning shows nested valid inferences across the four types. The results have implications for theoretical linguistics and English-language information processing, and the work identifies open questions about soundness and completeness of the AMI-1 foundation for future investigation, including formal meta-logical properties.
Abstract
This paper studies the validity and discourse reasoning of non-trivial generalized syllogisms involving the quantifiers in Square{most} and Square{all} from the perspective of knowledge reasoning. Firstly, this paper presents knowledge representations for these syllogisms and formally proves the validity of generalized syllogism AMI-1. Subsequently, 19 non-trivial generalized syllogisms, 22 non-trivial valid generalized modal syllogisms, 8 valid classical syllogisms, and 24 valid classical modal syllogisms are respectively deduced from the valid generalized syllogism AMI-1 on the basis of deductive reasoning. Additionally, this paper discusses how to judge the validity of discourse reasoning nested by the above four types of syllogisms, which have four types of figures and different forms. In conclusion, such formal deductions not only provide a theoretical foundation for English language information processing, but also provide methodological insights for studying other syllogistic systems.